Let T, S, and G represent train, social network, and games apps respectively. We need to select 5 apps such that there is at least one of each type.

Consider the following cases:

1. 1 train, 1 social, 3 games: The number of ways is given by:

   \(3 \times 6 \times \binom{14}{3} = 6552\)

2. 1 train, 3 social, 1 game: The number of ways is given by:

   \(3 \times \binom{6}{3} \times 14 = 840\)

3. 3 train, 1 social, 1 game: The number of ways is given by:

   \(\binom{3}{3} \times 6 \times 14 = 84\)

4. 2 train, 2 social, 1 game: The number of ways is given by:

   \(\binom{3}{2} \times \binom{6}{2} \times 14 = 630\)

5. 2 train, 1 social, 2 games: The number of ways is given by:

   \(\binom{3}{2} \times 6 \times \binom{14}{2} = 1638\)

6. 1 train, 2 social, 2 games: The number of ways is given by:

   \(3 \times \binom{6}{2} \times \binom{14}{2} = 4095\)

Summing all these possibilities gives the total number of ways:

\(6552 + 840 + 84 + 630 + 1638 + 4095 = 13839\)